follow the steps for graphing a rational function to graph the function f(x)=x + 23/x^3. determine the…

follow the steps for graphing a rational function to graph the function f(x)=x + 23/x^3. determine the behavior of the graph on either side of any vertical asymptotes, if any exist. select the correct choice and, if necessary, fill in the answer box(es) to complete your choice. a. it approaches ∞ on one side of the asymptote(s) at x = and -∞ on the other. it approaches either ∞ or -∞ on both sides of the asymptote(s) at x = (type integers or simplified fractions. use a comma to separate answers as needed. type each answer only once.) b. it approaches either ∞ or -∞ on both sides of the asymptote(s) at x = (type an integer or a simplified fraction. use a comma to separate answers as needed. type each answer only once.) c. it approaches ∞ on one side of the asymptote(s) at x = and -∞ on the other. (type an integer or a simplified fraction. use a comma to separate answers as needed. type each answer only once.) d. the function has no vertical asymptote. determine the horizontal asymptote(s), if any exist. select the correct choice and, if necessary, fill in the answer box(es) to complete your choice. a. the function has one horizontal asymptote, (type an equation. use integers or fractions for any numbers in the equation.) b. the function has two horizontal asymptotes. the top asymptote is and the bottom asymptote is (type equations. use integers or fractions for any numbers in the equations.) c. the function has no horizontal asymptote. determine the oblique asymptote(s), if any exist. select the correct choice and, if necessary, fill in the answer box(es) to complete your choice. a. the function has one oblique asymptote, (type an equation. use integers or fractions for any numbers in the equation.)

follow the steps for graphing a rational function to graph the function f(x)=x + 23/x^3. determine the behavior of the graph on either side of any vertical asymptotes, if any exist. select the correct choice and, if necessary, fill in the answer box(es) to complete your choice. a. it approaches ∞ on one side of the asymptote(s) at x = and -∞ on the other. it approaches either ∞ or -∞ on both sides of the asymptote(s) at x = (type integers or simplified fractions. use a comma to separate answers as needed. type each answer only once.) b. it approaches either ∞ or -∞ on both sides of the asymptote(s) at x = (type an integer or a simplified fraction. use a comma to separate answers as needed. type each answer only once.) c. it approaches ∞ on one side of the asymptote(s) at x = and -∞ on the other. (type an integer or a simplified fraction. use a comma to separate answers as needed. type each answer only once.) d. the function has no vertical asymptote. determine the horizontal asymptote(s), if any exist. select the correct choice and, if necessary, fill in the answer box(es) to complete your choice. a. the function has one horizontal asymptote, (type an equation. use integers or fractions for any numbers in the equation.) b. the function has two horizontal asymptotes. the top asymptote is and the bottom asymptote is (type equations. use integers or fractions for any numbers in the equations.) c. the function has no horizontal asymptote. determine the oblique asymptote(s), if any exist. select the correct choice and, if necessary, fill in the answer box(es) to complete your choice. a. the function has one oblique asymptote, (type an equation. use integers or fractions for any numbers in the equation.)

Answer

Explanation:

Step1: Find vertical asymptotes

Set the denominator of the rational - part equal to zero. For (f(x)=x + \frac{23}{x^{3}}), the denominator of the rational part is (x^{3}). Set (x^{3}=0), then (x = 0). As (x\to0^{+}), (\frac{23}{x^{3}}\to+\infty) and (f(x)=x+\frac{23}{x^{3}}\to+\infty). As (x\to0^{-}), (\frac{23}{x^{3}}\to-\infty) and (f(x)=x+\frac{23}{x^{3}}\to-\infty). So it approaches (\infty) on one side of the asymptote at (x = 0) and (-\infty) on the other.

Step2: Find horizontal asymptotes

The degree of the numerator of the rational - part ((23) can be thought of as (23x^{0})) is (0) and the degree of the denominator is (3). Also, considering the non - rational part (y = x). As (x\to\pm\infty), the term (\frac{23}{x^{3}}\to0), but the function (y=x+\frac{23}{x^{3}}) has no horizontal asymptote since the non - rational part (y = x) dominates as (x\to\pm\infty).

Step3: Find oblique asymptotes

Since (f(x)=x+\frac{23}{x^{3}}), as (x\to\pm\infty), the term (\frac{23}{x^{3}}\to0). The oblique asymptote is (y = x).

Answer:

For vertical asymptotes: C. It approaches (\infty) on one side of the asymptote(s) at (x = 0) and (-\infty) on the other. For horizontal asymptotes: C. The function has no horizontal asymptote. For oblique asymptotes: A. The function has one oblique asymptote (y=x)